Selmer groups for elliptic curves in ℤ l d -extensions of function fields of characteristic p

Andrea Bandini[1]; Ignazio Longhi[2]

  • [1] Università della Calabria Dipartimento di Matematica via P. Bucci - Cubo 30B 87036 Arcavacata di Rende (CS) (Italy)
  • [2] National Taiwan University Department of Mathematics N ∘  1 section 4 Roosevelt Road Taipei 106 (Taiwan)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 6, page 2301-2327
  • ISSN: 0373-0956

Abstract

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Let F be a function field of characteristic p > 0 , ℱ / F a ℤ l d -extension (for some prime l ≠ p ) and E / F a non-isotrivial elliptic curve. We study the behaviour of the r -parts of the Selmer groups ( r any prime) in the subextensions of ℱ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of ℱ / F .

How to cite

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Bandini, Andrea, and Longhi, Ignazio. "Selmer groups for elliptic curves in $\mathbb{Z}_l^d$-extensions of function fields of characteristic $p$." Annales de l’institut Fourier 59.6 (2009): 2301-2327. <http://eudml.org/doc/10455>.

@article{Bandini2009,
abstract = {Let $F$ be a function field of characteristic $p&gt;0$, $\mathcal\{F\}/F$ a $\mathbb\{Z\}_l^d$-extension (for some prime $l\ne p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of the $r$-parts of the Selmer groups ($r$ any prime) in the subextensions of $\mathcal\{F\}$ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of $\mathcal\{F\}/F$.},
affiliation = {Università della Calabria Dipartimento di Matematica via P. Bucci - Cubo 30B 87036 Arcavacata di Rende (CS) (Italy); National Taiwan University Department of Mathematics N ∘  1 section 4 Roosevelt Road Taipei 106 (Taiwan)},
author = {Bandini, Andrea, Longhi, Ignazio},
journal = {Annales de l’institut Fourier},
keywords = {Selmer groups; elliptic curves; function fields; Iwasawa theory},
language = {eng},
number = {6},
pages = {2301-2327},
publisher = {Association des Annales de l’institut Fourier},
title = {Selmer groups for elliptic curves in $\mathbb\{Z\}_l^d$-extensions of function fields of characteristic $p$},
url = {http://eudml.org/doc/10455},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Bandini, Andrea
AU - Longhi, Ignazio
TI - Selmer groups for elliptic curves in $\mathbb{Z}_l^d$-extensions of function fields of characteristic $p$
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2301
EP - 2327
AB - Let $F$ be a function field of characteristic $p&gt;0$, $\mathcal{F}/F$ a $\mathbb{Z}_l^d$-extension (for some prime $l\ne p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of the $r$-parts of the Selmer groups ($r$ any prime) in the subextensions of $\mathcal{F}$ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of $\mathcal{F}/F$.
LA - eng
KW - Selmer groups; elliptic curves; function fields; Iwasawa theory
UR - http://eudml.org/doc/10455
ER -

References

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  1. P. N. Balister, S. Howson, Note on Nakayama’s lemma for compact Λ -modules, Asian J. Math. 1 (1997), 224-229 Zbl0904.16019MR1491983
  2. A. Bandini, I. Longhi, Control theorems for elliptic curves over function fields, Int. J. Number Theory 5 (2009), 229-256 Zbl1234.11071MR2502807
  3. A. Bandini, I. Longhi, S. Vigni, Torsion points on elliptic curves over function fields and a theorem of Igusa Zbl1247.11083
  4. Jordan S. Ellenberg, Selmer groups and Mordell-Weil groups of elliptic curves over towers of function fields, Compos. Math. 142 (2006), 1215-1230 Zbl1106.11021MR2264662
  5. Lisa A. Fastenberg, Mordell-Weil groups in procyclic extensions of a function field, Duke Math. J. 89 (1997), 217-224 Zbl0903.14006MR1460621
  6. Ralph Greenberg, Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997) 1716 (1999), 51-144, Springer, Berlin Zbl0946.11027MR1754686
  7. Ralph Greenberg, Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry (Park City, UT, 1999) 9 (2001), 407-464, Amer. Math. Soc., Providence, RI Zbl1002.11048MR1860044
  8. A. Grothendieck, Revêtements étales et groupe fondamental (SGA 1), (2003), Société Mathématique de France, Paris MR2017446
  9. Jun-ichi Igusa, Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves, Amer. J. Math. 81 (1959), 453-476 Zbl0115.38904MR104669
  10. Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183-266 Zbl0245.14015MR444670
  11. J. S. Milne, Étale cohomology, 33 (1980), Princeton University Press, Princeton, N.J. Zbl0433.14012MR559531
  12. J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) (1986), 167-212, Springer, New York Zbl0604.14018MR861976
  13. Jürgen Neukirch, Algebraic number theory, 322 (1999), Springer-Verlag, Berlin Zbl0956.11021MR1697859
  14. Jürgen Neukirch, Alexander Schmidt, Kay Wingberg, Cohomology of number fields, 323 (2000), Springer-Verlag, Berlin Zbl0948.11001MR1737196
  15. Tadashi Ochiai, Fabien Trihan, On the Selmer groups of abelian varieties over function fields of characteristic p &gt; 0 , Math. Proc. Cambridge Philos. Soc. 146 (2009), 23-43 Zbl1156.14037MR2461865
  16. Ambrus Pál, Proof of an exceptional zero conjecture for elliptic curves over function fields, Math. Z. 254 (2006), 461-483 Zbl1196.11090MR2244360
  17. Jean-Pierre Serre, Local fields, 67 (1979), Springer-Verlag, New York Zbl0423.12016MR554237
  18. Tetsuji Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), 415-432 Zbl0602.14033MR833362
  19. Tetsuji Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), 211-240 Zbl0725.14017MR1081832
  20. Joseph H. Silverman, The arithmetic of elliptic curves, 106 (1986), Springer-Verlag, New York Zbl0585.14026MR817210
  21. Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, 151 (1994), Springer-Verlag, New York Zbl0911.14015MR1312368
  22. Joseph H. Silverman, The rank of elliptic surfaces in unramified abelian towers, J. Reine Angew. Math. 577 (2004), 153-169 Zbl1105.11016MR2108217
  23. Fabien Trihan, On the Iwasawa Main Conjecture of abelian varieties over function fields of characteristic p &gt; 0  Zbl1301.11056
  24. Douglas Ulmer, Elliptic curves with large rank over function fields, Ann. of Math. (2) 155 (2002), 295-315 Zbl1109.11314MR1888802
  25. Douglas Ulmer, Jacobi sums, Fermat Jacobians, and ranks of abelian varieties over towers of function fields, Math. Res. Lett. 14 (2007), 453-467 Zbl1127.14021MR2318649

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